Anchored burning bijections on finite and infinite graphs

Abstract

Let $G$ be an infinite graph such that each tree in the wired uniform spanning forest on $G$ has one end almost surely. On such graphs $G$, we give a family of continuous, measure preserving, almost one-to-one mappings from the wired spanning forest on $G$ to recurrent sandpiles on $G$, that we call anchored burning bijections. In the special case of ${\mathbb{Z}}^d$, $d\ge 2$, we show how the anchored bijection, combined with Wilson's stacks of arrows construction, as well as other known results on spanning trees, yields a power law upper bound on the rate of convergence to the sandpile measure along any exhaustion of ${\mathbb{Z}}^d$. We discuss some open problems related to these findings.

26 pages; 1 EPS figure. Minor alterations made after comments from referee

References in corpus (1)

• Rate of convergence estimates for the zero dissipation limit in Abelian sandpiles

Cited by in corpus (1)

• Asymptotic height distribution in high-dimensional sandpiles